Exponents and Logarithms Quiz

Questions and Answers

What are the objectives of learning regarding exponents and logarithms?

Evaluate ax in various cases, simplify algebraic expressions using exponent rules, evaluate logarithms, and solve equations using logarithm rules.

In the expression M = bn, what is referred to as the exponent?

According to Rule 1 of exponents, what is the result of am × an?

am+n

What is the value of 20?

1 Signup and view all the answers

What should b be in a logarithm to ensure M is positive?

b must be positive Signup and view all the answers

According to Rule 4 of logarithms, what is loga a equal to?

1 Signup and view all the answers

The general form of logarithms is log b M = ___

n Signup and view all the answers

What is log2 1 equal to?

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What is the answer to 2^x = 16?

x = 4 Signup and view all the answers

What is the solution to the logarithmic equation log2 40 + 3 log2 6 - log2 135?

6 Signup and view all the answers

What is the answer to the equation 25(10)^(2t) = 208?

t = 0.46 Signup and view all the answers

Study Notes

Exponents

Exponential form is represented as ( M = b^n ) where ( b ) is the base and ( n ) is the exponent.
Rules of Exponents:
- Rule 1: ( a^m \times a^n = a^{m+n} )
- Rule 2: ( \frac{a^m}{a^n} = a^{m-n} )
- Rule 3: ( (a^m)^n = a^{mn} )
- Rule 4: ( (ab)^m = a^m \times b^m )
- Rule 5: ( \sqrt[n]{a^m} = a^{\frac{m}{n}} )
- Rule 6: ( a^0 = 1 )
- Rule 7: ( a^{-m} = \frac{1}{a^m} )

Simplifying Expressions

Examples to simplify:
- ( \frac{x^4 \times x^4}{x^2} ) results in ( x^{6} )
- Result of ( (x^{-1} y^{3})^{2} = \frac{y^{6}}{x^{2}} )

Solving Exponential Equations

Solve ( 2^x = 16 ) leading to ( x = 4 )
Solve ( \frac{a^3}{2} = 25 ) and find ( a )

Logarithms

General form: ( \log_b M = n ) indicates ( b^n = M ).
Logarithmic functions are inverses of exponential functions.
Common logarithms (base 10) often omit the base; other bases should be indicated.

Rules of Logarithms

Rule 1: ( \log_b (a \times c) = \log_b a + \log_b c )
Rule 2: ( \log_b \left( \frac{a}{c} \right) = \log_b a - \log_b c )
Rule 3: ( \log_b (a^n) = n \log_b a )
Rule 4: ( \log_a a = 1 )
Rule 5: ( \log_a 1 = 0 )
Rule 6: ( b^{\log_b r} = r )
Rule 7: ( \log_m b = \frac{\log_n b}{\log_n m} )

Example Problems with Logarithms

Simplify ( \log_2 40 + 3 \log_2 6 - \log_2 135 ) resulting in ( 6 ).
Solve ( 2^x = 7 ) and find ( x \approx 2.807 ).
Solve ( 25^{x-1} = 3^x ) leading to ( x \approx 0.293 ).

Additional Practice

Additional problems include solving equations of the form:
- ( 25(10)^{2t} = 208 )
- ( 38 + 12e^{-0.5t} = 208 )
- ( \log_{10} x + \log_{10} (x+9) = 1 )
Aim to practice simplifying without calculators for improved skills.

Motivational Quote

"Don't say you don't have enough time. You have the same number of hours per day as historical figures who achieved greatness."

Description

Test your understanding of exponents and logarithms with this quiz. Explore the key rules and properties that govern exponential notation and logarithmic functions. Focus on the relationship between exponents and logarithms to enhance your mathematical skills.